#9724829

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,14)(13,17)(15,19)(16,23)(20,24)(21,22)(25,26)(27,33)(28,32)(29,31)(30,34)(35,39)(36,38)(37,40);(2,5)(6,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27,30,29)(26,31,34,33)(28,36,37,35)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,14,19,17)(11,20,23,21)(16,22,18,24)(25,26,30,34)(27,31,29,33)(28,32,37,40)(35,38,36,39);(1,3,4,8)(2,6,5,7)(9,13,15,12)(10,17,19,14)(11,21,23,20)(16,24,18,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,11)(10,18)(12,20)(13,21)(14,24)(15,23)(16,19)(17,22)(25,28,30,37)(26,32,34,40)(27,35,29,36)(31,38,33,39);(9,14,15,17)(10,12,19,13)(11,22,23,24)(16,20,18,21)(25,29,30,27)(26,33,34,31)(28,35,37,36)(32,38,40,39)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9632025);